The two major points of difference between European soccer and American sports are the competition structure and how player transactions work. Anything else is comparatively minor and if it comes up, I'll talk about it then (you may also feel free to ask me, of course).
Unlike in the US (where it's just the league championship or bust), soccer teams have multiple chances to win competitions every season. The important ones are as follows:
play is the major one and most games played are league games. Each country has its own league system. There are no conferences or divisions or the like within the leagues: every team plays every other team twice - one home and one away - and at the end of the season the team with the most points wins the league title. If you finish in the bottom couple of positions, you get relegated to a lower league; likewise, if you finish in the top couple of positions, you get promoted to a higher league (if applicable). There's no postseason tournament, but some lower leagues have a mini playoff to secure a final promotion spot.
- Domestic cup
. Less prestigious than league play, at least for the big boys, this is a season-long knockout tournament that runs concurrently with the league season and is open to pretty much any professional or semi-professional team in the country. You could be having a garbage season in the league and still save your job by going on a nice cup run, and a trophy is a trophy. Every country has at least one cup, though they very in importance. England and France have two; I'll go into the specifics and differences if they become relevant.
- The Champions League
. Despite the league name, this is also a season-long tournament and is the most prestigious competition in Europe. If you finish in the top couple of spots in your country's top league, you get to play in the Champions League the next year along with the best teams from the other countries. The Champions League Final, held in late May or early June, is always the final event in every (club) season and is Europe's version of the Super Bowl.
- The Europa League
. Similar to the Champions League, for teams who finished just a bit below
Champions League contention every year, or were otherwise knocked out of the Champions League at a certain point. Despite similarities to the much-derided NIT, the Europa League is generally treated with respect unless you're from England.
- Several minor one-or-two-off tournaments that you're entered in if you won stuff the previous season. Not worth going into detail about here.
- International tournaments
: Club teams are not national teams! The World Cup is every four years. Every continent also has its own continental tournament; Europe's and South America's are the most competitive. Most continental tournaments run every four years unless they're in the process of realigning the dates (South America). North America's runs every two years. I will generally not be reporting on these in much detail beyond talking about if my players had a good tournament or something.
Whenever a player switches teams, it's called a transfer
. Most transfers come in the form of Team A paying Team B an amount of money for the right to negotiate a new contract with the player. Player-for-player trades are comparatively rare, and losing a player to free agency (known as a free transfer) is looked at with fear, because it means you weren't able to get any money for that player.
Player preference plays a huge part in this. If you send a player to a new team, they have to agree to contract terms with that new team before the deal can happen. In other words, you can't exile a player to Siberia F.C. because he was annoying you and you wanted to send a message.
Soccer, in general, revolves around money, moreso than even the American sports. There is no salary cap beyond what your club can afford to spend (technically this isn't true but we won't get into that here). If you're not a top team, you're incentivized to sell your better players if they want to leave - better to get something for them before they join a better team when their contract expires and leave you with nothing. You then use that cash to buy replacements, hope you go on a good run, and the cycle renews itself.
There is zero incentive to tank. If your team has a crappy season, the result is you being relegated to a lower league and losing out on money you get for being in a better league. You don't even get the satisfaction of getting a high pick in a player draft: all new players come up through your own youth system and are randomly generated based on a number of factors that include how much you've invested in it and how good that country generally is at soccer.
Some top teams, in no particular order: